Definitions: A set is an unordered collection of elements. When \(A\) and \(B\) are sets, \(A = B\) (set equality) means \[\forall x ( x\in A \leftrightarrow x \in B)\] When \(A\) and \(B\) are sets, \(A \subseteq B\) (“\(A\) is a subset of \(B\)") means \[\forall x (x \in A \to x \in B)\] When \(A\) and \(B\) are sets, \(A \subsetneq B\) (“\(A\) is a proper subset of \(B\)") means \[(A\subseteq B) \wedge (A \neq B)\]